Parabolic quasilinear equations minimizing linear growth functionals
type: Item
instance of: scholarly article
title: Parabolic quasilinear equations minimizing linear growth functionals
zbMATH Open document ID: 1053.35002
author: Fuensanta Andreu-Vaillo
author: V. Caselles
author: J. M. Mazón
published in: Progress in Mathematics
publication date: 2004-03-08T00:00:00Z
review text: This book is devoted to PDE's of elliptic and parabolic type associated to functionals having a linear growth in the gradient, with a special emphasis on the applications related to image restoration and nonlinear filters. The main model is the total variation functional (i.e. the \(L^1\) norm of the gradient, in a \(C^1\) setting) proposed in a seminal paper by \textit{L. Rudin, S. Osher} and \textit{E. Fatemi} in 1992 [cf. Physica D 60, No. 1--4, 259--268 (1992; Zbl 0780.49028)]. Chapter 1 contains a review of the variational approach to image restoration based on the total variation minimization, with a description of the Euler-Lagrange equation and of the constraints involved. It contains also some numerical computations and a review of the main numerical methods adopted in the literature. Chapter 2 deals with the parabolic initial value problem \[ \begin{aligned} \frac{\partial u}{\partial t}= \text{div}\biggl( {Du\over | Du| }\biggr)&\quad \text{in }(0,\infty)\times\Omega,\\ u(0,x)=u_0(x)& \quad \text{in }\Omega\end{aligned} \] with Neumann boundary conditions, in a bounded domain \(\Omega\) of \({\mathbb R}^n\). The main result is the existence and the uniqueness of a weak solution for any initial datum in \(L^1\). Here ``weak'' refers mainly to the fact that \(D u/| D u| \) is not everywhere defined, so that even distributional solutions have to be properly defined. The proof is mainly based on the Crandall-Liggett generation theorem. Chapter 3 deals, instead, with the same problem in \(\Omega={\mathbb R}^n\). After an initial discussion about the existence of strong solutions when the initial datum is in \(L^2({\mathbb R}^n)\), the main result is the existence, uniqueness and stability of entropy solutions (in the sense first introduced by Kruzhkov for scalar conservation laws) when the initial data are in \(L^1_{\text{loc}}({\mathbb R}^n)\). Also the case of measures for initial data is discussed. Chapter 4 is devoted to the asymptotic properties of soluti…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 49K10
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 94A08
zbMATH DE Number: 2052486
zbMATH Keywords: total variation minimization
zbMATH Keywords: BV functions
zbMATH Keywords: image processing
zbMATH Keywords: entropy solutions
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2052486